Weighted restriction theorems for space curves (Q996909)

Let \(W(L^p,l^q)(\mathbb R^n)\) be an amalgam space with the norm \[ \| f\| _{W(L^p,l^q)}=\left(\sum_{k\in\mathbb Z^n}\left(\int_{[0,1)^n+k}| f(x)| ^p\,dx\right)^{q/p}\right)^{1/q} \] if \(1\leq p,q<\infty\) (and with usual modification when \(p\) or \(q\) is infinite). There are three main results in this paper. The first main result of the paper under review is the following Theorem 1.1. Let \(\gamma(t)\) be a non-degenerate \(C^n\) curve in \(\mathbb R^n,\) \(t\in I,\) where \(I\) is a finite interval (a typical example of such curve is \((t,t^2,\dots,t^n)\)). Let \(1\leq p<1+2/(n^2+n)\) and \(1/p+[2/(n^2+n)]/q\geq1.\) Suppose that \(\nu\) is a nonnegative (measurable) function such that \(\nu^{-1/(p-1)}\in W(L^1,l^\infty).\) Then there exists a constant \(C,\) independent of \(f\) and \(\nu\) and depending only on \(\gamma,\) \(n,\) and \(I,\) such that \[ \left(\int_I | \widehat f(\gamma(t))| ^q\,dt\right)^{1/q}\leq C\left\| \nu^{-1/(p-1)}\right\| ^{1-1/p}_{W(L^1,l^\infty)}\left(\int_{\mathbb R^n}| f(x)| ^p\nu(x)\,dx\right)^{1/p}. \] Here, as usual, \(\widehat f\) is the Fourier transform of \(f.\) Then this result is generalized to a class of oscillatory integral operators related to restriction estimates. In the last theorem restriction estimates for curves with two weight functions are established.